High Complexity Tilings with Sparse Errors

Abstract

Tile sets and tilings of the plane appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). The idea is to enforce some global properties (of the entire tiling) by means of local rules (for neighbor tiles). A fundamental question: Can local rules enforce a complex (highly irregular) structure of a tiling?The minimal (and weak) notion of irregularity is aperiodicity. R. Berger constructed a tile set such that every tiling is aperiodic. Though Berger’s tilings are not periodic, they are very regular in an intuitive sense.In [3] a stronger result was proven: There exists a tile set such that all n×n squares in all tilings have Kolmogorov complexity Ω(n), i.e., contain Ω(n) bits of information. Such a tiling cannot be periodic or even computable.In the present paper we apply the fixed-point argument from [5] to give a new construction of a tile set that enforces high Kolmogorov complexity tilings (thus providing an alternative proof of the results of [3]). The new construction is quite flexible, and we use it to prove a much stronger result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.